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An Improved Upper Bound on the Capacity of the Primitive Diamond Relay Channel

Deniz Gündüz, Yanxiao Liu, Yi Liu

arXiv:2610.12034v1cs.ITmath.IT

TL;DR

The primitive diamond relay channel capacity is unknown in general, and the cut-set bound may be loose. This paper adapts a strengthened-converse method to derive a single-letter upper bound. The bound coincides with the strengthened Gaussian bound and can be strictly tighter than existing bounds for selected discrete memoryless channels.

  • Problem

    The primitive diamond relay channel capacity is unknown in general, and the cut-set bound can be loose because it permits unrestricted cooperation across a cut.

    Source: B. Upper and Lower Bounds on the… · p. 2

  • Method

    The paper adapts the Csiszár–Körner–Marton sum identity and Gallager-type identification to derive a single-letter upper bound for the two-relay channel.

    Source: B. Upper and Lower Bounds on the… · p. 2, §II · p. 3

  • Results

    The proposed bound coincides with the strengthened Gaussian upper bound and, for selected discrete memoryless channels, is strictly tighter than both existing upper bounds.

    Source: §III · p. 5, §V · p. 7

  • Takeaways & Limitations

    The proposed bound provides a tighter outer bound for selected discrete memoryless primitive diamond relay channels than the cut-set bound and the existing upper bound.

    Source: §V · p. 7

    Directly introducing auxiliary receivers did not improve the outer bound in the authors’ investigation.

    Source: §V · p. 7

Abstract

from arXiv · show

We study the capacity of the primitive diamond relay channel and derive an improved upper bound compared with existing upper bounds by employing the Csisz\'ar--K\"orner--Marton sum identity and Gallager-type identification. For Gaussian primitive diamond relay channels, we show that our proposed bound coincides with the existing strengthened Gaussian upper bound of Wu, \"Ozg\"ur, Peleg, and Shamai. For discrete memoryless channels, our bound can be strictly tighter than both the cut-set bound and the existing upper bound, and we assess the improvement using different channel instances, together with a decode--forward/compress--forward time-sharing inner bound.

A. Channel Model

This section introduces the primitive diamond relay communication system and states a channel power constraint.

A. Channel Model

  • The channel is presented as a primitive diamond relay communication system.Source: Figure 1 · p. 1
  • The channel input is subject to the average power constraint E[X2] ≤P.Source: A. Channel Model · p. 1

B. Upper and Lower Bounds on the Capacity

The capacity of the primitive diamond relay channel is unknown in general, and the paper develops a stronger converse by adapting an existing method. It also uses a DF–CF time-sharing scheme as an achievable benchmark.

B. Upper and Lower Bounds on the Capacity

  • The primitive diamond relay channel capacity remains unknown, and the cut-set bound is a natural but generally not tight upper bound.The cut-set bound permits unrestricted cooperation across a cut, motivating a converse that accounts for dependence among the source input, relay observations, and relay descriptions.Source: B. Upper and Lower Bounds on the… · p. 2
  • Adapting the Csiszár–Körner–Marton sum identity and Gallager-type identification converse for the primitive relay channel yields a new upper bound for the two-relay model.The cited primitive-relay converse gives a single-letter characterization; its Gaussian bound coincides with, while its discrete bound can be strictly tighter.Source: B. Upper and Lower Bounds on the… · p. 2
  • The benchmark time-shares decode-and-forward and distributed compress-and-forward/noisy network coding, jointly optimizing the time-sharing fraction and relay-link capacity allocation.For Gaussian channels, it uses Gaussian inputs and independent Gaussian quantization test channels.Source: B. Upper and Lower Bounds on the… · p. 2

II. THE NEW UPPER BOUND

The new converse gives a single-letter upper bound using relay-specific auxiliary variables. Its bound is at least as tight as the cut-set bound for general primitive diamond relay channels.

II. THE NEW UPPER BOUND

  • Theorem 5 gives an upper bound on every achievable rate for a primitive diamond relay channel.Source: §II · p. 3
  • The bound ranges over distributions p(x)p(y1|x)p(y2|x)p(u|x, y1)p(v|x, y2), with U and V conditionally tied to their respective relay observations given X.The auxiliary variables can be restricted to |U| ≤ |X||Y1| + 1 and |V| ≤ |X||Y2| + 1.Source: §II · pp. 3–4
  • The converse proof uses the Csiszár–Körner–Marton sum identity and a time-sharing variable to obtain single-letter constraints.It defines U and V from time-indexed quantities and the time-sharing variable Q.Source: §II · p. 3
  • By non-negativity of mutual information terms, Theorem 5 is at least as tight as the cut-set bound for general primitive diamond relay channels.Source: §II · p. 5

III. GAUSSIAN PRIMITIVE DIAMOND RELAY CHANNEL

The section develops a Gaussian upper bound for the primitive diamond relay channel and shows that it coincides with the bound in Theorem 2. The proof evaluates the bound using Gaussian distributions and reduces the optimization to a restricted parameterization.

III. GAUSSIAN PRIMITIVE DIAMOND RELAY CHANNEL

  • The equivalence proof combines a rotation trick, trigonometric substitution, and explicit optimization steps.The substituted constraints are related to those of Theorem 2, with explicit optimization used for the remaining terms.Source: §III · pp. 5–6
  • Because the bounds (6a)–(6e) depend on ρ and ρ̂ only through their squares, attention may be restricted to ρ, ρ̂ ∈ [0, 1].Source: §III · p. 5
  • The upper bounds in Theorem 2 and Theorem 5 coincide for the Gaussian primitive diamond relay channel.Theorem 5’s Gaussian evaluation suffices with Gaussian distributions for PX, PX,Y1|U, and PX,Y2|V.Source: §III · p. 5
  • The derived constraint includes R ≤ C1 + C2 + log sin ψ1 + log sin ψ2.The constraint appears among the transformed bounds used to establish equivalence with Theorem 2.Source: §III · p. 6

IV. EVALUATION FOR DISCRETE CHANNELS

The evaluation compares the proposed upper bound with the cut-set bound, the discrete upper bound of, and a DF–CF time-sharing inner bound across three discrete channel instances.

IV. EVALUATION FOR DISCRETE CHANNELS

  • The comparisons use C1 = C2 = C and include the cut-set bound, the upper bound of, and the DF–CF time-sharing inner bound.The section considers multiple channel instances to assess the new upper bound.Source: §IV · p. 7
  • For the symmetric instance, both marginal channels are BSC(0.3), with the comparison shown in Figure 2a.Source: §IV · p. 7
  • For the asymmetric instance, the marginal channels are BEC(0.2) and BSC(0.1), with the comparison shown in Figure 2b.Source: §IV · p. 7
  • For the degraded instance, TY1|X is BSC(0.3) and TY2|X cascades BSC(0.3) with BEC(0.1), with bounds shown in Figure 2c.Source: §IV · p. 7

V. CONCLUDING REMARKS

The paper derives a single-letter upper bound and reports that it is strictly tighter than both existing upper bounds for selected discrete memoryless channels. Its Gaussian specialization coincides with ’s strengthened bound; direct auxiliary-receiver introduction did not improve the bound.

V. CONCLUDING REMARKS

  • For selected discrete memoryless channels, the proposed bound is strictly tighter than both existing upper bounds, with a visible numerical gap.The comparisons include the cut-set bound and the upper bound of.Source: §V · p. 7
  • The Gaussian specialization coincides with the strengthened Gaussian upper bound of.Source: §V · p. 7
  • Directly introducing auxiliary receivers did not improve the outer bound; refined applications remain a proposed direction for future work.The gap between known inner and outer bounds motivates seeking a tighter outer bound.Source: §V · p. 7

AI STATEMENT

The authors disclose AI assistance across literature searches, simulations, proof development, and manuscript drafting, while taking responsibility for accuracy and integrity.

AI STATEMENT

  • AI systems assisted with literature searches, numerical simulations, inner-bound exploration, proofs of Lemmas 1 and 2, and manuscript drafting.The authors state that they take full responsibility for the manuscript’s accuracy and integrity.Source: AI STATEMENT · p. 7

APPENDIX

The appendix establishes that jointly Gaussian distributions suffice to evaluate the Gaussian-channel upper bound and proves convexity of the set used in its optimization. These results support the min-max exchange used in the bound's derivation.

APPENDIX

  • For the Gaussian channel, jointly Gaussian distributions suffice to evaluate the upper bound, with Gaussian optimizers for both relay-conditioned distributions.The argument establishes Gaussian optimality for PX,Y1|U and, symmetrically, PX,Y2|V.Source: THE PROOF OF LEMMA 1 · pp. 8, 11
  • The set A is convex, enabling the min-max exchange used in the upper-bound derivation.The proof constructs a mixture distribution that preserves the Markov-chain conditions and power constraint.Source: THE PROOF OF LEMMA 1 · p. 8
  • The Gaussian-optimality proof uses independent copies and the Skitovich–Darmois theorem to establish Gaussianity of the input optimizer.Equality forces transformed outputs to be independent, which implies independence of transformed inputs and then Gaussianity of X.Source: THE PROOF OF LEMMA 1 · p. 10
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